Editing Logistic distribution (section)

{{short description|Continuous probability distribution}}
{{Probability distribution|
  name       =Logistic distribution|
  type       =density|
  pdf_image  =[[File:Logisticpdfunction.svg|320px|Standard logistic PDF]]|
  cdf_image  =[[File:Logistic cdf.svg|320px|Standard logistic CDF]]|
  parameters =<math>\mu,</math> [[location parameter|location]] ([[real number|real]])<br /><math>s > 0,</math> [[scale parameter|scale]] (real)|
  support    =<math>x \in (-\infty, \infty)</math>|
  pdf        =<math>\frac{e^{-(x-\mu)/s}} {s\left(1+e^{-(x-\mu)/s}\right)^2}</math>|
  cdf        =<math>\frac{1}{1+e^{-(x-\mu)/s}} = \frac{1 + \tanh \frac{x-\mu}{2s}}{2}</math>|
  quantile   =<math>\mu+s \log\left(\frac{p}{1-p}\right)</math>|
  mean       =<math>\mu</math>|
  median     =<math>\mu</math>|
  mode       =<math>\mu</math>|
  variance   =<math>\frac{s^2 \pi^2}{3}</math>|
  skewness   =<math>0</math>|
  kurtosis   =<math>6/5</math>|
  entropy    =<math>\ln s + 2</math>|
  mgf        =<math>e^{\mu t}\Beta(1-st, 1+st)</math><br />for <math>t \in (-1/s,1/s)</math><br />and <math>\Beta</math> is the [[Beta function]]|
  char       =<math>e^{it\mu}\frac{\pi st}{\sinh(\pi st)}</math>
| ES         =<math>\mu + \frac{sH(p)}{1-p}</math> <br /> where <math>H(p)</math> is the binary entropy function<ref name="norton">{{cite journal |last1=Norton |first1=Matthew |last2=Khokhlov |first2=Valentyn |last3=Uryasev |first3=Stan |year=2019 |title=Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation |journal=Annals of Operations Research |volume=299 |issue=1–2 |pages=1281–1315 |publisher=Springer|doi=10.1007/s10479-019-03373-1 |arxiv=1811.11301 |url=http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf |access-date=2023-02-27 |url-status=dead |archive-url=https://web.archive.org/web/20230301065519/http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf |archive-date= Mar 1, 2023 }}</ref> <math>H(p) = -p \ln(p) - (1-p) \ln (1-p)</math>

}}

In [[probability theory]] and [[statistics]], the '''logistic distribution''' is a [[continuous probability distribution]]. Its [[cumulative distribution function]] is the [[logistic function]], which appears in [[logistic regression]] and [[feedforward neural network]]s. It resembles the [[normal distribution]] in shape but has heavier tails (higher [[kurtosis]]).  The logistic distribution is a special case of the [[Tukey lambda distribution]].